Tools

"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or sub-elliptic geometry, as well as on graphs and to certain non-local Sobolev norms. It only uses elementary cut-off argu ..."

We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or sub-elliptic geometry, as well as on graphs and to certain non-local Sobolev norms. It only uses elementary cut-off arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,

... the limiting case of a Trudinger inequality. We are not claiming that the type of argument above is new. Indeed, very similar (if not identical) ideas have been used recently by various authors. See =-=[2, 3, 6, 8, 13, 16, 17]-=- among others. In particular, our work was stimulated by the results of G. Carron [6]. After this paper was written we received the preprint [12] which also uses similar argument. However, it seems to...

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the co ..."

In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word &quot;Gaussian&quot; we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...

"... We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."

We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

"... We give upper estimates on the long time behaviour of the heat kernel on a non-compact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."

We give upper estimates on the long time behaviour of the heat kernel on a non-compact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.

...he heat kernel long time behaviour is closely related to the FaberKrahn inequalities - lower estimates ofs1(\Omega\Gamma via the volume (\Omega\Gamma1 This was established in the case of manifolds in =-=[6]-=- and [29], and for the case of graphs in [16, Proposition V.1]. Proposition 2.3 Assume that v 0 := inf x2\Gamma (x) ? 0: (2.11) Suppose also that, for all non-empty finite sets\Omega , one hass1(\Omeg...

"... Let M be a Riemannian manifold, and ∆ be the Laplace-Beltrami operator on M. It is known that there exists a unique minimal positive fundamental solution to the associated heat equation, which is referred to as the heat kernel and denoted by pt(x, y) (x, y ∈ M, t> 0). ..."

Let M be a Riemannian manifold, and ∆ be the Laplace-Beltrami operator on M. It is known that there exists a unique minimal positive fundamental solution to the associated heat equation, which is referred to as the heat kernel and denoted by pt(x, y) (x, y ∈ M, t&gt; 0).

...at the uniform upper bounds of the heat kernel are closely related to isoperimetric type inequalities including the Sobolev’s, Nash’s and the logarithmic Sobolev inequalities. More recent works [G2], =-=[Carr]-=-, [C2] revealed the importance of a Faber-Krahn type inequality and of a generalized Nash inequality (see also the surveys [G4] and [C3]). The situation is quite different with lower bounds of the hea...

...ned out that isoperimetric inequalities are in close connection to ultra-contractive semigroup theory, particularly to diagonal uppers estimates of transition probability of random walks( [96], [97], =-=[17]-=-, [39]). All these works demonstrate that functional analytic inequalities (like Sobolev, Nash or Poincaré inequality ) and isoperimetric inequalities are also in strong connection. Let us mention her...